Papers

In recent years, Krylov space approaches have emerged as a useful framework for characterizing dynamical chaos in a wide variety of physical systems. However, the connection between Krylov dynamics and the established structures of classical chaos remains largely unexplored. Classical chaos is characterized by exponential stretching and mixing. For quantum maps with a classical limit, we show that these two signatures are encoded in different structures within Krylov space: stretching is encoded in the geometry of the Krylov operators as determined by the Lyapunov exponent, while mixing manifests itself as an exponential decay of the Arnoldi sequences in agreement with the Ruelle-Pollicott resonances.

Learning with data in the form of quantum states is of current interest and has led to a variety of problems that boil down to interaction with the available data via quantum measurement and classical post-processing of observed classical outcomes. In quantum measurement PAC learning, one is given a sequence of unknown, prepared quantum states and classical labels, along with a hypothesis class of candidate measurements. The task is to select a measurement from the hypothesis class that minimizes a fixed notion of error in prediction of the classical labels via measurement of a new state by the selected hypothesis. In this work, we consider the advantage of interacting with the given data in the measurement learning framework using learning rules given by measurements that cannot be implemented using local operations and classical communication (LOCC), as opposed to single-copy learning rules. We provide a construction showing that there exist learning scenarios wherein single-copy learning rules are asymptotically suboptimal compared to optimal ones. We then show that learning rules based on entangled measurements enjoy at most a polynomial sample complexity advantage over single-copy learning rules in the PAC learning setting (under a natural joint measurability covering assumption).

We construct a dimension-dependent family of generalized equiangular measurements that gives explicit steering witnesses in every prime dimension $d\geq5$. The construction consists of $d-1$ mutually unbiased measurements and an additional normalized family of $2d-1$ Hermitian operators depending on a parameter $\tau$. Positivity of the latter family is the main technical problem. We show that the noncentral operators reduce to three-dimensional invariant blocks, while the remaining central operator is exactly solvable in a two-dimensional subspace. Two unitarily equivalent $3\times3$ matrix representations describe the same active spectrum, but yield different Gershgorin enclosures because Gershgorin bounds are basis dependent. Combining the stronger endpoint from each representation gives a fully analytic asymmetric interval for $\tau$ that guarantees a valid POVM. Combining this POVM with the $d-1$ MUMs yields an explicit GEAM. Finally, using a maximally entangled realization and the local-hidden-state bound previously derived for generalized equiangular measurements, we obtain an explicit steering inequality and a sufficient criterion guaranteeing quantum violation in every prime dimension $d\geq5$.

Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL-states and QL-gate operations, emerging from the dynamics of complex synchronized networks. The present work builds on this framework to present a simulation of quantum dynamics in spin systems. We focus on the transverse field Ising model, a widely studied Hamiltonian for interacting spin systems. We show how circuits, built using the QL analog of quantum gates, can be used to prepare an arbitrary initial wavepacket and simulate the time evolution of the resulting state for extracting physical observables such as magnetization. Beyond the statevector dynamics, we introduce a set of metrics that characterize how QL-circuit operations transform the underlying graph connections. These metrics quantify the microscopic coupling, phase coherence, and effective inter-block coupling generated by the circuit. These graph-level quantities provide constraints and design targets for potential physical implementations, indicating where interconnection resources, phase control, and effective coupling strength must be engineered in a hardware realization. Overall, this study demonstrates how classical resources can be harnessed to simulate quantum systems and we discuss the advantages and limitations.

The 'unreasonable effectiveness of mathematics in the natural sciences' may at times give the impression that natural science, especially modern theoretical physics, is all but a formal apparatus. However, a key challenge of physics remains to align everyday notions of experience – be it directly or mediated through instruments of arbitrary sophistication – with their mathematical formalisation. This applies in particular to quantum theory, arguably the most successful theory of physics ever invented. One of its core assumptions postulates: $Observables\ correspond\ with\ self-adjoint\ operators.$ Here, we argue that already this most basic premise about the quantum formalism falls apart under careful scrutiny, at least when understood for all observables at once. In other words, we challenge the role of the noncommutative algebra of bounded operators as an adequate formalisation of the 'algebra of observables', understood as (equivalence classes of) measurement procedures that admit a coherent empirical interpretation.

The Minimum Vertex Cover (MVC) problem is a fundamental NP-hard combinatorial optimization problem with applications in network analysis and resource allocation. Grover's algorithm provides a quadratic reduction in query complexity for unstructured search, but existing Grover-based MVC formulations can incur substantial quantum resource overhead due to costly vertex-counting circuits and complex oracle constructions. We develop and evaluate several encoding and oracle-design strategies for reducing the qubit count, circuit depth, and gate complexity of Grover-based MVC search. First, we construct a Dicke-Parallel formulation that restricts the search to fixed-cardinality subsets, eliminating explicit vertex counting, together with a parallel edge-verification oracle that reduces feasibility-checking overhead. We then develop an Edge-Counting formulation that replaces per-edge auxiliary storage with a logarithmic-size counting register, substantially reducing ancillary-qubit requirements. Finally, we propose an Edge-Centric encoding that represents endpoint selections directly and derives vertex-selection states through incident-edge Boolean operations, enabling more depth-efficient oracle construction. Our resource analysis reveals complementary trade-offs among qubit width, circuit depth, gate count, and Grover iteration count. Edge-Counting is particularly attractive under tight qubit constraints, while Edge-Centric can reduce both circuit depth and Grover iteration count when the graph has a moderate edge count and sufficient representation multiplicity. Dicke-Parallel provides a more robust choice when such multiplicity is limited or graph density makes the edge-based search space large. These results provide practical guidance for selecting Grover-based MVC formulations according to hardware constraints and graph structure.

GHZ-based conference key agreement distributes a shared key to $N$ parties in one network use, whereas pairwise BB84 needs $N-1$. The survival of that advantage has been studied against party number, loss, and topology, but almost always with the channel fixed at depolarizing. Simulating both protocols exactly for $N = 3, 4, 5$ under dephasing, amplitude damping, and depolarizing noise, we find the crossover is set by the channel's Pauli error composition, not its identity: normalized to average infidelity, amplitude damping and depolarizing agree to within about 1% at each group size, while dephasing can tolerate roughly $2.4\times$ more at $N = 3$. We derive a closed-form rate for arbitrary per-party Pauli noise and use it to prove that concentrating a fixed noise budget on one arm is strictly worse than spreading it, and that giving the dealer role to the cleanest arm improves the rate by roughly 18%. One structural fact drives every result: the phase-error term is a product over all $N$ arms, while the bit-error term stays local to a pair.

Identity impersonation fraud may emerge only after digital account opening, requiring an early post-onboarding approach based on behavioral and transactional signals observed shortly after activation. Confirmed cases remain rare, creating extreme class imbalance and material operational risk for digital banks. We evaluate quantum machine learning (QML) against classical one-class methods in this setting using a real 2025 cohort of 3,419 digital account openings from Banco de Creditos e Inversiones (BCI), Chile, including 8 confirmed impersonation-fraud cases (0.23% prevalence). The data were provided through an industry-research collaboration between BCI and CoreDevX LABTAV under operational constraints. The best-performing configuration used only 2 qubits and was executed on CoreDevX's SpinQ Triangulum II, a 3-qubit nuclear magnetic resonance (NMR) device, enabling hardware validation beyond classical simulation. The quantum one-class classifier based on Automatic Quantum Feature Mapping (AQFM) achieves 88% recall (7 of 8 frauds detected) and 9.21% precision, corresponding to approximately 70% fewer false positives than the best-performing classical baseline at comparable recall. When its 2025-calibrated decision threshold is applied unchanged to the independent 2026 cohort, AQFM detects 20 of 31 frauds (64.5% recall) with 8.81% precision and remains the best-performing model among those evaluated.

We investigate how photon loss and environmental memory affect probe selection and measurement sensitivity for estimating the dispersive coupling strength in a cavity qubit system. Using density operator and quantum state diffusion approaches, we examine the quantum Fisher information for representative separable and entangled initial states in Markovian and non Markovian regimes. The results show that closed system generator variance alone does not determine open system performance. The survival of parameter sensitive coherences and the influence of environmental memory also play important roles. Within the single photon preparations examined, a separable probe can outperform the entangled family considered. Comparisons of selected coherent and squeezed vacuum preparations at equal mean photon number further illustrate how photon number statistics and qubit preparation affect sensitivity. For the parameters studied, finite environmental memory can enhance and prolong sensitivity and yield nonzero QFI for a probe whose Markovian QFI vanishes. We also numerically identify a fixed Hermitian observable whose error propagation uncertainty remains close to the quantum Cramer Rao bound over a prescribed coupling interval in the Markovian regime. These findings clarify the interplay between probe preparation, environmental dynamics, and measurement choice in dispersive quantum metrology.

Variational autoencoders are powerful representation learning models that map complex data into low-dimensional latent spaces, enabling the discovery of interpretable and disentangled factors. Such representations can facilitate the interpretation and controllable generation of data describing complex scientific systems. Understanding how these factors are organized and encoded in latent space is therefore important for developing reliable representation learning models. Recently, quantum variational autoencoders (QVAEs) have been proposed as quantum representation models, demonstrating informative latent representations and improved latent-space occupancy through quantum regularization. However, it remains unclear whether and how QVAEs can learn disentangled and interpretable latent factors. A key challenge in investigating quantum latent factors is that a small number of qubits spans an exponentially large Hilbert space, making the notion of an individual quantum latent dimension nontrivial. Here, we investigate what constitutes an individual quantum latent dimension and whether it can encode a distinct factor. We develop theoretical insights into quantum latent dimensions and support them with empirical studies on representative synthetic problems, including MNIST variants. Across three datasets, we demonstrate that QVAEs can discover factorized and semantically interpretable latent representations, with individual qubits functioning as meaningful latent factors. These results establish a foundation for understanding quantum latent spaces and their potential for structured and interpretable representation learning.

Generalized entanglement wedges (GWs), proposed by Bousso and Penington for gravitating regions in general spacetimes, share many properties of ordinary entanglement wedges, including nesting, monotonicity and strong subadditivity of their generalized entropies. In AdS/CFT, the corresponding properties of entanglement wedges are consequences of the structure of the dual CFT. In this paper we ask whether the properties of GWs have a similar origin. We explore this question in vacuum AdS$_3$ at leading order in $G$. We first show that every GW on a static slice is a union of geodesically convex regions. Each such region is an intersection of ordinary entanglement wedges, so every GW is built from unions and intersections of entanglement wedges. We then use the Crofton formula of integral geometry to write the perimeter of a convex region as the measure of the set of geodesics that intersect it. In AdS$_3$ every geodesic is the Ryu–Takayanagi surface of a boundary interval, so this measure can be written in terms of the von Neumann entropies of CFT intervals. Strong subadditivity of the CFT entropy guarantees that the measure is positive. Monotonicity and strong subadditivity of GWs then follow from simple inequalities between sets of geodesics. The strong subadditivity of GWs is therefore a direct consequence of strong subadditivity on the boundary. Our results further suggest that GWs correspond to subsystems of the CFT that are more general than boundary subregions.

Designing expressive quantum circuits for the variational quantum eigensolver (VQE) is a central challenge in simulating interacting many-body systems. Even symmetry-preserving (SP) circuits need not access the full many-body symmetry sector. We demonstrate this explicitly for the spin-$1/2$ XXZ and Hubbard models, where Hamiltonian-inspired nearest-neighbor (NN) SP circuits generate restricted dynamical Lie algebras (DLAs). While these restricted manifolds can accurately represent the ground state near the free-fermion limits, $J_z=0$ for XXZ and $U=0$ for Hubbard, interactions expose inaccessible many-body directions and severely limit VQE performance even at large circuit depth. Remarkably, a minimal modification of the generator connectivity can reconstruct the full orthogonal DLA within all relevant symmetry sectors studied in this work. For XXZ model, the addition of only one SP next-to-NN single-excitation connection triggers this dramatic reconstruction, whereas the Hubbard case requires a SP correlated double-excitation;–showing that the relevant connectivity can extend from the spatial lattice to many-body configuration space. This reconstruction enlarges the tangent space and accessibility of the energy-lowering Hamiltonian direction, leading to near-unit ground-state fidelity in VQE simulations. Our results establish a direct link between generator connectivity and variational accessibility and suggest a broader circuit-design principle: identify minimal symmetry-preserving generators that enlarge the accessible many-body space rather than increasing circuit depth alone. This perspective may have broader relevance for quantum many-body simulation at the interface of condensed-matter physics and quantum computing.

Quantum magic is a necessary resource for fault-tolerant quantum computing, capturing the departure of quantum states from the classically simulable stabiliser set. Motivated by its role in quantum information processing, we investigate the correlation structure of states that maximize the Rényi stabiliser entropies, focusing on the T-type and Hoggar states. Inspired by their distinctive properties, we explore the notion of Pauli flat states, defined as N-qubit states whose expectation values on all Pauli strings are equal in absolute value. These states reproduce the correlation structure of highly magical states, which could also be interpreted as a mixed variant of Pauli covariant N-qubit symmetric informationally complete (SIC) Weyl-Heisenberg fiducial states, while not necessarily possessing nontrivial quantum magic. We investigate their existence and possible constructions, including examples both inside and outside of the stabiliser polytope. To identify the latter, we examine a magic witness tailored to the introduced family of states. Finally, we discuss their entanglement and Bell nonlocality, providing a broader characterization of their quantum properties. For completeness, we find pure states with a relaxed condition on the Pauli expectations, which forces unequal elements to zero, and discuss their non-stabiliserness. Our results offer a new perspective on the relation between quantum magic and Pauli correlation structures. They hold relevance to geometrical approaches to quantum mechanics, entanglement and magic detection, as well as estimation tasks.

Interpreting neutrino oscillation measurements requires computing the probability that a neutrino born with one flavor is detected with another. In vacuum and in matter of constant density, this probability has exact formulas. Where the density varies along the path, as in the Earth, the Sun, or a supernova, it does not. There, the usual approaches either replace the profile by constant-density steps, whose error falls only as the square of their width, or integrate the evolution equation with a general-purpose solver, which must shorten its steps to meet a tighter tolerance. Both become slow at high accuracy, yet an analysis needs the probability at many energies, directions, and parameter values. Each new experiment or model often requires writing and validating a new solver. We present Magnus, an open-source Python code that avoids these limitations. It uses the Magnus expansion, whose error falls as a high power of its step width and whose cost is set by how fast the density varies, not by how many oscillations the path holds. Magnus accepts any density profile, any number of flavors, and any Hermitian Hamiltonian, such as those of non-standard interactions, sterile neutrinos, Lorentz-invariance violation, and pseudo-Dirac neutrinos. Where a cheaper route exists, it takes it automatically. For example, where the density varies slowly, as in the Sun, it follows the instantaneous eigenstates and applies the expansion only at resonances. Besides oscillating probabilities, Magnus returns phase-averaged ones, which describe solar and astrophysical neutrinos. A typical probability takes a few milliseconds, and batched scans are one to two orders of magnitude cheaper per point. Among the public codes we compared, Magnus is the fastest at high accuracy on a varying profile. Magnus shifts the effort of studying a new experiment or model from writing a solver to specifying its physics.

Floquet topological order (FTO) extends topology beyond equilibrium, giving rise to phenomena, such as the dynamical transmutation of anyons. Here, we realize $\mathbb{Z}_N$ FTO in a generalized Kitaev honeycomb model with clock degrees of freedom using a three-step periodic drive. The driven system admits an emergent description in terms of constrained parafermions coupled to conserved $\mathbb{Z}_N$ gauge fields. At the boundary, the drive generates chiral parafermionic Floquet edge modes. We resolve their propagation through correlation functions, characterize their fractional quantum-information transport through the radical chiral unitary index, and use their dynamics to probe exchange statistics. In the bulk, we explore the dynamical anyon transmutation with a non-equilibrium loop order parameter. At a Floquet sweet spot, where the drive becomes Clifford, these phenomena can be accessed using stabilizer simulations. Short-time matrix-product-state simulations away from this point show that the chiral edge response persists at accessible times. Our results establish $\mathbb{Z}_N$ Kitaev models as a setting for exploring intrinsically non-equilibrium topological phenomena on qudit-based quantum processors.

Exponentially small spectral gaps are the central obstacle to adiabatic quantum annealing, yet whether such bottlenecks demand correspondingly complex entanglement remains unclear. We show that gap scaling and entanglement complexity need not be linked. Conventional first-order transitions combine exponentially small gaps with bounded $O(1)$ bipartite Schmidt rank, even though the critical states retain global multipartite correlations, while continuous transitions combine polynomially closing gaps with logarithmically growing entanglement. We show that each pairing can also be inverted: in a known first-order transition in a staggered-field antiferromagnetic Ising chain the gap closes polynomially despite bounded Schmidt rank, while we construct a mirror-twin manifold model whose gap closes exponentially even as its entanglement entropy grows logarithmically with system size. We derive bounds connecting Schmidt rank to Hamming-space geometry, and identify manifold density and connectivity as the ingredients that determine which combination occurs. These results show that entanglement complexity alone cannot characterize annealing difficulty, and point toward a broader search for quantum resources that capture the geometry of state evolution in Hilbert space.

We construct a family of chaotic brickwork quantum circuits exhibiting superdiffusive transport with dynamical exponent $z=3/2$ in the full Kardar–Parisi–Zhang (KPZ) universality class. This is enabled by the equilibrium current associated with a inhomogeneous conserved density having a nontrivial dependence on the chemical potential. Using nonlinear fluctuating hydrodynamics, we derive the propagation velocity, broadening scale, and universal scaling form of the charge correlation function from microscopic equilibrium properties. Tensor-network simulations of a two-site spin-$1$ (qutrit) circuit reproduce these parameter-free predictions, including the full stationary KPZ profile. We further construct a three-site spin-$\frac12$ (qubit) circuit, where transport varies from diffusive to superdiffusive with the chemical potential, while long-lived coherent quasiparticles delay the convergence of the profile to the KPZ scaling form. Our results establish that KPZ superdiffusion in quantum systems is not restricted to fine-tuned integrable models.

Bosonic quantum error correction represents a route towards significant hardware overhead reduction for fault-tolerant quantum computation. However, controlling quantum harmonic oscillators that encode the quantum information comes with its own challenges. In many platforms such as superconducting circuits and trapped ions, the bosonic degree of freedom is controlled by an ancillary qubit whose decay during a qubit-oscillator gate propagates to the oscillator as a continuous error. Such errors generally lie beyond the scope of errors that the bosonic code is capable of correcting. We introduce the continuous-variable flag: an auxiliary oscillator coupled to the ancilla whose phase-space position encodes the continuous error corrupting the storage mode. After the gate, a heterodyne measurement of the flag estimates this continuous-variable error and a subsequent feedback control undoes it up to a bosonic code stabilizer. The scheme requires no new ingredients, as the flag is driven by the same dispersive coupling, displacements and rotations that implement the gates it is supposed to protect. We construct sequences protecting conditional rotations to all orders in ancilla decay with one flag, and conditional displacements to first order with one flag and to all orders with two. Simulations with realistic circuit-QED parameters predict more than three orders of magnitude fewer logical errors for the parity measurement of a four-component cat code, and a finite-energy GKP logical lifetime near its noiseless-ancilla bound. By discretizing continuous-variable errors, the flags introduced in this work are a useful tool for any hybrid oscillator-qubit processor, and open the route towards fault-tolerant simulation and computation with bosonic modes.

We prove optimal query bounds for recovering a single qubit from black-hole radiation in the Haar random oracle model, when the remaining black hole contains at most one sixteenth of the system's qubits. Recovery with any constant Haar-averaged advantage over trivial decoding requires queries proportional to the Hilbert-space dimension of the remaining black hole. The lower bound is unconditional for decoders chosen independently of the sampled unitary, allows arbitrary computation between queries, and access to $U, U^\dagger, U^*, U^\mathsf T$ along with the controlled variants. An existing decoder using only forward and inverse queries attains a matching bound. The proof uses a path recording oracle to compare real and maximally mixed states. As applications, we obtain an efficiently preparable, statistically far, computationally indistinguishable (EFI) pair and quantum commitments relative to a public Haar oracle, as well as prove a tight linear rank lower bound for Uhlmann transformation on a fixed-target family.

Quantum batteries store energy in nonequilibrium quantum states. The ergotropy sets the work extractable by an arbitrary global unitary, but a device usually exchanges energy through a single local port. It remains unclear how much stored work such a port can deliver, and whether topological edge modes help. We study a dimerized spin chain of Su-Schrieffer-Heeger (SSH) type. Excited two-level ancillas charge it at one end by repeated collisions, and an initially empty two-level load discharges it at the same site. An exact free-fermion mapping yields a bound on the load work for all number-conserving quadratic extraction protocols. The load gains extractable work only if a single-particle mode of the battery holds more than half an excitation. Resonant charging fills the topological edge mode above this threshold. A single boundary pulse then delivers 98.6% of the maximal load work allowed by the bound for representative parameters. Under the same six-collision controls, the trivial-phase load remains passive. Within the protocol that selects the load pulse for maximum delivered work, cycle power peaks at a finite collision number while delivered work continues to increase over the scanned range. Over the tested range, the advantage survives hopping disorder, which preserves chiral symmetry. On-site disorder breaks this symmetry and leaves a finite fraction of loads passive. Topology thus supplies a localized access channel, while spectral matching and symmetry decide whether stored energy becomes usable local work.

Braiding and fusing anyons are fundamental to topological quantum computation, yet existing circuit protocols often employ ancillas and measurements. We determine the minimal circuit resources for performing such anyon operations in quantum double models of generic finite groups. Our approach braids anyons through a sequential unitary circuit without ancillas. Intuitively, this is obtained by deforming the underlying graph on which the anyons live. More generally, we introduce graph rewiring, a diagrammatic calculus that assigns a geometric interpretation to individual two-qudit gates and tracks their action on stabilizers. For fusion, we prove a general no-go theorem excluding purely unitary protocols for fusing unknown anyon types without introducing ancillas. Remarkably, we show such "input-agnostic" fusion is enabled by simply using a qudit reset. Hence, contrary to folklore, ancilla resources are not needed to braid or fuse non-Abelian anyons (non-Abelions). This work has implications in experimental efforts in error correction: our fusion protocol highlights reset as an underutilized primitive for error correction without the spacetime-overhead of ancillas and measurements.

The measures of distance frequently used in quantum information, such as trace distance or fidelity-based purified distance, are the smallest extensions of the corresponding distance measures in classical information theory. However, this way of extending classical distances is not unique, and a whole spectrum of other extensions can be defined. Here we study the question of how quantum information processing tasks are affected by a change of the error metrics to the maximal extensions of the classical distance measures, imposing more demanding error requirements on operational tasks. We find that the performance of some tasks, such as channel coding or quantum resource distillation, is not affected whatsoever by this change. However, for other tasks such as channel simulation and quantum resource dilution, we show that the rates are now given not by functions of the standard (Umegaki) quantum relative entropy, but ones based on the Belavkin-Staszewski relative entropy. Our results connect the Belavkin-Staszewski relative entropy with operational tasks and reveal that known reversibility results in quantum information, including the reverse Shannon theorem and the asymptotic reversibility of quantum resource transformations, no longer hold under maximal distance measures.

We investigate converse bounds on the average decoding error probability in finite-blocklength classical-quantum channel coding. We first present a lower bound for multiple quantum hypothesis testing in terms of pairwise trace distances and derive a corresponding fidelity bound. We then obtain a spectrum-based converse that depends only on the a priori probabilities and spectra of the states. We show that the converse bound remains tight for the quantum depolarizing channel under suitable conditions. For codes with product-state outputs, this converse takes an explicit form involving products of output-state eigenvalues. We apply it to binary codes over the quantum amplitude damping channel using the input states $\vert+\rangle$ and $\vert-\rangle$. For this setting, we also discuss a normal approximation to the spectrum-based converse in the large blocklength regime. In all numerical examples considered, the spectrum-based converse is tighter than the other converse bounds at low noise levels.

An important question in quantum information theory asks whether every gate in the Clifford hierarchy is semi-Clifford or not, as such gates admit resource-efficient implementations via gate teleportation. Extending recent results for prime dimensions~\cite{silva_clifford_2025}, we prove that every hierarchy gate on a single qudit of dimension \(d\) is semi-Clifford, if and only if \(d\) is square-free. In composite dimensions, \(\mathbb{Z}_d^2\) is a symplectic module rather than a vector space, motivating a distinction between four types of gates. Semi-Clifford gates ($\mathcal{SC}$) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class $\mathcal{N}$ admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates ($\mathcal{LSC}$) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates ($\mathcal{GSC}$) map some maximal abelian Pauli subalgebra to another under conjugation. In square-free dimensions, we have \(\mathcal{SC}=\mathcal{LSC}\) and \(\mathcal{N}=\mathcal{GSC}\), but when \(d\) is not square-free, these equivalences can fail, as Lagrangian submodules can become non-free. We also show that every third-level gate of the one-qudit Clifford hierarchy is a generalized semi-Clifford in arbitrary dimensions.

Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting $Q(f\circ g)=\Theta(Q(f)Q(g))$ for all Boolean functions $f$ and $g$; a direct sum theorem asserting that computing $k$ copies of a function (or search problem) costs $\Omega(k)$ times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires $\Omega(k)$ times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution $\mu$ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing $k$ copies of $f$ when the input comes from the product distribution $\mu^k$ instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new “multiplicative” variant of the $\gamma_2$ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new “Shaltiel-free” measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.

We determine the two-way quantum, entanglement-distribution, private, and secret-key capacities of the pure-loss bosonic channel under an unconditional mean transmitted-photon-number constraint. For transmissivity $\eta$ and mean photon number $N$, all four capacities coincide and are given by $g(N)-g((1-\eta)N)$, where $g$ is the entropy of a thermal mode. This resolves the longstanding energy-constrained capacity problem by establishing the optimality of the reverse-coherent-information rate introduced in 2009. Our central tool is sector teleportation simulation, which we develop specifically to transfer the input-energy constraint to the shared entanglement resource, thereby preserving the constraint throughout converse arguments for general adaptive protocols. For every $N>0$, we further show that the corresponding strong-converse thresholds coincide with the unconstrained value $-\log_2(1-\eta)$. For parallel protocols with a hard total-photon-number cutoff, the same energy-constrained capacity formula instead satisfies an exponential strong converse. As an extension, we determine the energy-constrained two-way capacities of the quantum-limited amplifier, showing that classical assistance does not improve its unassisted quantum and private rates. Together, these results establish a unified finite-energy framework for adaptive bosonic communication and reveal how energy constraints fundamentally reshape both attainable rates and converse bounds.

Magic state cultivation is a promising method for preparing high-fidelity non-Clifford resource states - a central component of fault-tolerant quantum computing - with low spacetime overhead. In benchmarking cultivation, existing resource estimates assume access to noiseless transversal measurement of the output magic state to allow for postselection during the final escape stage of the cultivation process. However, such measurements would destroy the magic state and are therefore unavailable during computation. We develop an open-boundary postselection method, Caliper, that uses only nondestructive mid-circuit syndrome information and substantially outperforms existing methods operating under these constraints. In some regimes, our postselection method uniquely recovers the resource efficiency of prior estimates. However, in other cases, we observe a tradeoff. At spacetime costs reported in previous works, the achievable logical error rates can be orders of magnitude higher, and recovering the reported logical error rate requires substantially larger codes and longer escape stages. Our results highlight how practical considerations involving mid-circuit postselection can impact the design and performance of magic state cultivation protocols.

We identify a barrier that helps explain why proving stronger quantum state-preparation lower bounds has been so difficult. In particular, we establish a quantum analogue of the Razborov-Rudich natural proofs barrier for state-preparation lower bounds. We call a property of quantum states natural if it holds for a sufficiently large fraction of Haar-random states and can be efficiently tested when given all of the state's amplitudes. Under a standard cryptographic assumption, we show that no natural property can prove superpolynomial state-preparation lower bounds even against a fixed level of the Magic Hierarchy. We show that several existing state-preparation lower-bound techniques are natural in our sense, including arguments based on approximate degree, not being a unique ground state of a local Hamiltonian, and mutual information.

Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width $f$, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance $d$ and check weight $w$. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations. First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by $1$ guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an $[[1088,128,22]]$ LP code with $f=22$ and an $[[1024,256,24]]$ PP code with $f \leq 27$, both at $w=10$. At $w=11$, our PP constructions attain either $d=25$ with $f \leq 27$ or $d=24$ with the proved optimum $f=25$, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.

We prove that every pure state of stabilizer rank at most $k$ has stabilizer extent at most $2^{O(\sqrt{k\log(k+1)})}$, and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-$k$ operators. This implies stabilizer fidelity at least $2^{-O(\sqrt{k\log(k+1)})}$, resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an $\Omega(n^2/\log n)$ lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of $T$-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states. As a consequence of the Clifford rank–norm inequality, we obtain an $\Omega(n^2/\log n)$ lower bound for exact representations of $n$-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank $O((\log n)^2/\log\log n)$, respectively, a $\log n$ improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most $k$, with $poly(n)2^{O(\sqrt{k\log(k+1)})}$ time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.